Suppose I have probabilities matrix between 3 states, for exampel we can take $P=\left(\begin{array}{ccc} \frac{1}{9} & \frac{8}{9} & 0\\ 0 & 0.3 & 0.7\\ 0 & 0 & 1 \end{array}\right)$, and we want to find the stationary probability. In order to do so we solve the equlation $wP=w$ where w is the vector of stationary distribution. Here $w=\left(\begin{array}{ccc} 0 & 0 & 0 \end{array}\right)$. Does it mean the matrix converges to a zero matrix? Does it have any special meaning ? can we make any consequences about the process (except there isn't any absorbing state of course)?
2026-04-24 18:16:06.1777054566
Can a stationary distribution be zero vector
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Well first off probability matrices have rows that sum up to one. So your example is not a probability matrix. Markov chains are used to describe a physical process where a system evolves with time. If the zero vector is the solution, then the physical significance is that the "mass" completely escapes as time goes to infinite.